Questions tagged [bivariate-distributions]

For questions on bivariate distributions, the combined probability distribution of two randomly different variables.

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Marginal density equal to zero everywhere.

I've been working on a two part question on bivariate transformations and marginal densities but am having difficulty finding where I have made a mistake as the final answer for the marginal density ...
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Show that U and V are independent.

Suppose we have (X, Y ) be a bivariate normal vector with mean vector µ = (0, 0) and covariance matrix $$ Σ= \begin{bmatrix}2 & 1 \\1 & 2 \\\end{bmatrix} $$ I'm given: $$ A^TΣA=\begin{bmatrix}...
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Indepedent copy of Bivariate Normal

In this question, Correlated joint normal distribution: calculating a probability Most upvoted answer obtained independent copy using this equation, $\pmatrix{U\\V}=\Sigma^{-1/2} \pmatrix{X\\Y}$. I ...
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The bivariate normal distribution and its ellipse

Let $f(x,y)$ be a bivariate normal p.d.f. and let $c$ be a positive constant so that $c < \left(2\pi \sigma_{X} \sigma_{Y}\sqrt{1-\rho^2}\right)^{-1}$. Show that $c=f(x,y)$ defines an ellipse in ...
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Single integral over bivariate normal distribution

I have $(X, Y)$ are bivariate normal with $\mu_X=log(160), \mu_Y =log(165),\sigma_x=0.05=\sigma_y$, and $\rho=0.5$. After a change of variables, I want the marginal density $f_U(u) = \int_{0}^\infty\...
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Using general bivariate gaussian to extract marginal PDF from given bivariate PDF

I had a homework question to find the marginal probability density functions, $p_X(x)$ and $p_Y(y)$, given a join probability density function $p_{XY}(x,y)$. I have solved the problem by integration i....
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Distribution of a bivariate Gaussian

Let $X,Y$ be a bivariate normal distribution with correlation $\rho$, means $\mu_1 \mu_2$ and variances $\sigma_1^2, \sigma_2^2$. Is there an expression for $$P(X \ge 0, Y \ge 0)$$ that only uses $\...
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Determining the marginal density from joint probability

Given joint probability $$P(x,y) = \frac{1}{8\pi}e^{{-a(bx^2-cxy+dy^2)}}$$ where $a,b,c,d$ are rational fractions How would I go about determining the marginal probability of $p(x)$ and $p(y)$? I feel ...
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Mixture of Independent Bi-variate Gaussian distributions admits two different interpretations with different results

Here is the problem I am trying to solve and understand. Suppose I have a factorized joint distribution $q(v^1,v^2)=q(v^1)q(v^2)$ from which I can easily sample from. This distribution gives the mean ...
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Confusing in limits of bivarite distribution and contradiction in related question

This question is from "Mathematical Statistics with application by Wackerly and Mendelhall" $8th$ edition page $233$ ,question $5.5$ : $f(y_1,y_2)=3y_1 , 0 \leq y_2 \leq y_1 \leq1$ and $0$ ,...
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confusion on determining integral limits in bivarite distribution

$f(x_1,x_2)= 8x_1x_2 , 0<x_1<x_2<1 $ $\text{and}$ $0 , \text{elsewhere}$ I want to find $E(X_1X_2^2).$ According to the book: $$E(X_1X_2^2) = \int_{0}^{1}\int_{0}^{x_2} (x_1x_2^28x_1x_2)...
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Pdf of sum of independent rvs is the convolution of pdfs proof

I am trying to prove the statement in the title. However, I want some help to make my derivation mathematically rigorous. I have $X_{1}$ and $X_{2}$ which are two independent rvs. In addition, I have $...
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Covariance for a bivariate normal distribution

I have a question concerning the bivariate normal distribution where I have been able to prove that the first covariance we are asked for is equal to $0$, but was not sure if this would be implying ...
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Find $a$ such that the $P(X \in s) = 0.95$ where $ s = \{(x_1,x_2)^T \in \Bbb R : -a \leq x_1 \leq a \} $. Standard bivariate normal distribution.

I have a bivariate normal distribution, where $ X = (X_1, X_2)^T$. The means are given by $\mu = (0,0)^T$ and the covariance matrix is $ \sum = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} ...
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Prove Independence of a Bivariate Normal Distribution

Two variables $X,Y$ are bivariate normally distributed. We know that $Var(X)=Var(Y)$. Show that the two random variables $X+Y$ and $X-Y$ are independent. I'm feeling pretty stumped by this question, ...
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Solution to bivariate normally distributed problem

Is anyone able to solve the following integral with a normal bivariate PDF, or at least to clarify if there exist a closed form solution? Thanks in advance. \begin{equation} \frac{1}{2\pi} \int_{-\...
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Given probability density $f(x, y)=12xy(1-y)$, find the joint density of $U=XY^2$ and $V=Y$ and the marginals

Let $X$ and $Y$ be random variables with probability density $f(x, y)=12xy(1-y)$ if $0<x<1, 0<y<1$. Find the joint probability density of $U=XY^2$ and $V=Y$. Find the marginal density of $...
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Find marginal probability using bivariate normal

I'm given two variables $\begin{bmatrix} X_{1}\\{X_2} \end{bmatrix} \sim N_2({\mu}=\begin{bmatrix} 2\\-5\end{bmatrix},\Sigma = \begin{bmatrix} 1&{-0.5}\\{-0.5}&4 \end{bmatrix})$ How would I ...
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Trinomial and bivariate normal distribution - checking answers.

please could someone check through my answers for this question. Thanks! X,Y,Z follow a trinomial distribution with success probabilities $p_X, p_Y, p_Z > 0$ and $p_X + p_Y + p_Z = 1$ and sample ...
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Expected Value of X,Y from Covariance Matrix

Let $(X,Y)$ be a bivariate random variable with a Gaussian distribution on $\mathbb{R}^2$, mean zero and variance-covariance matrix: $$C=\begin{pmatrix} 0.42 & -0.42\\-0.42 & 0.42\end{pmatrix}$...
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Does Covariance equal 0

Let $U,V$ be a bivariate random variable with a continuous distribution and $f_{U,V}$ is the joint density of $(U,V)$. Suppose that $f_{U,V}(−u,v)=f_{U,V}(u,v)$ for all $u,v∈\mathbb{R}$, then $cov(U,V)...
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Expectation of $|x^{2} - y^{2}|$ when $x$ and $y$ are bivariate normal

I was asked the following question by my professor. Suppose $x$ and $y$ are jointly normal variables. We further suppose the marginal distribution of $x$ and $y$ are $N(0, 1)$ and $N(0, 2)$ ...
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Finding max of $var(a^TY)$.

Let $X=(X_1,X_2,X_3)^T$ be a multivariate random variable with the standard Gaussian distribution on $\mathbb{R}^3$. Define the multivariate random variable $Y=(Y_1,Y_2,Y_3)^T$ by $$\begin{pmatrix} ...
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131 views

Deriving the joint probability distribution of a transformed rv

Suppose that A and B are 2 r.v with joint probability distribution given by $f_{AB}(a,b) = -\frac{1}{2}(ln(a)+ln(b))$, if $0 \lt a \lt 1$ and $0 \lt b \lt 1$, and 0 otherwise. Define $X = A+B$ and $Y =...
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Finding the probability of a bivariate joint random variable

Let $A=(M,N)^T$ be a bivariate random variable with joint density defined by $$f_{M,N}(m,n) = \frac{3}{2 \pi} \sqrt{m^2+n^2}$$ if $m^2+n^2<1$ and $0$ otherwise. Let $B = (F,G)^T$ be given by $$\...
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Absolute value of expected value

Given $(A_1,A_2)$ to be the bivariate random variable in $\mathbb{R}^2$ with mean $0$ and cov($A_1,A_2)$ $= -1.05$ and $Var(Z_1) = Var(Z_2) = 1.05$. How do I find the expected value of $E(|A_1A_2|)$? ...
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Finding covariance matrix from a 3-variate RV

Let M be a 3-variate standard Gaussian distribution on $\mathbb{R}^3$ given by $M=(M_1,M_2,M_3)^T$ .Let $V=(V_1,V_2)^T$ be defined by $$V= \begin{pmatrix}1.1 & M_3\\0 & 1.1\end{pmatrix}\begin{...
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Finding the expected value of a bivariate gaussian distribution

Suppose that $(A,B)$ have a standard Gaussian distribution on $\mathbb{R}^2$. How do I find the expected value for $\mathbb{E}[max(3.9A+B,A+3.9B)]$? I know that A and B follow the standard normal ...
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1 answer
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Proof that $\mathbb{E}\bigl[1-\Phi(x_1 - \theta_1) \mid \theta_2 \leq x_2-σx_1+σθ_1\bigr]$ is increasing in $\sigma$

Numerically it appears that the following function is increasing (or at least non-decreasing) in $σ$, $$ f(x_1,x_2;σ)=∫_{-∞}^{∞}∫_{-∞}^{x_2-σx_1+σθ_1}[1-Φ(x_1-θ_1)]φ(θ_1)φ(θ_2)dθ_2dθ_1 $$ where $\phi$ ...
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Finding $P(X \leq a | Y = k)$ where $X\sim\operatorname{Exp}(\lambda)$ and $Y = [X]$ where $[\cdot]$ rounds up to the nearest integer.

$Y = [X]$ where $[\cdot]$ rounds up to the nearest integer. It's given that $X \sim \operatorname{Exp}(\lambda)$. The first part asks me to show that $Y \sim \operatorname{Geo}(1-e^{-\lambda})$. The ...
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Guessing a function that maximises the variance?

I'm doing a question that discusses a six-sided fair die being rolled and then discussing the numbers that appear on the top and on the side facing you. The rest of the question discusses the ...
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How is the variance of a variable in terms of variance of other variable?

I want to transform the variables $(x,y)$ on the bivariate normal distribution. $$ p(x,y) = \frac{1}{2\pi \sigma_x \sigma_y (1-\rho^2)^{1/2}} \exp\left[ -\frac{1}{2(1-\rho^2)^{1/2}}\left( \frac{x^2}{\...
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Bivariate Normal Distribution and sub vector

Let $$(X,Y)\sim N(\mu^\rightarrow ,\Sigma)$$ we cannot assume anything about the dependancy between X and Y. Can we assume the following? $$X\sim N(\mu_x,\sigma ^2_x)~,~Y\sim N(\mu_y, \sigma^2_y)$$...
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how to use correlation to calculate condition expectation

Suppose (X, Y) is Bivariate Normal, with X,Y ~ N(0, 1), and Corr(X, Y) = $\rho$ , please find E(Y|X) and E(X|Y). I don't have a clue how to solve this, anyone could give me a pointer? Just for the ...
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What is the Bayesian classifier given two bivariate Gaussians with same covariance matrix?

Given two bivariate Gaussian distributions, and conditional distributions of feature vector $X=(X_1,X_2)$, given label $Y = y$, given by \begin{equation} (X_1, X_2) \sim \left\{ \begin{array}{...
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Calculate covariance of two bivariate normal random variables [closed]

I'm reading Modern Mathematical Statistics with Applications and I don't know how to get the covariance of $\rho$ from $X$ and $Y$ in Example 6.16: Where do I start?
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What does it mean for a joint PDF to be symmetric in its constituents?

Let $X$ and $Y$ be independent standard normal random variables and consider the following linear transformations, $$U = X$$ and $$V = \rho X + \sqrt{1 - \rho^2}Y,$$ where $\rho \in (0, 1)$. Show that ...
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Proving that uncorrelatedness implies independence for bivariate joint normal random variables

Question Let $X$ and $Y$ be independent standard normal random variables and consider the following linear transformations, $$U = aX + bY$$ and $$V = cX + dY,$$ where $a, b, c, d \in \mathbb{R}$. Find ...
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Finding the distribution of X + Y using mean variance and correlation coefficient

Suppose $(X,Y)$ has the bivariate normal distribution with mean $(0,1)$ and variance $(1,1)$ and correlation coefficient $.5$. What is the distribution of $X + Y$? Now, I am in an accelerated ...
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Can any continuous function of two variables be described as a joint of two continuous functions?

(1) Would this be a valid definition of a joint function $f$ of two functions $g$ and $h$? $$g: \mathbb{X} \rightarrow \mathbb{R};\,\,\, h: \mathbb{Y} \rightarrow \mathbb{R}$$ $$f: \mathbb{X} , \...
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In search of a bivariate distribution with nicely behaved integrals over a surface

I am looking for a joint distribution with infinite support on $R^2$, where $P(Y>max(c,aX+b))$, where $a$,$b$ and $c$ are real numbers, has a closed-form solution (without integrals). I tried many ...
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Gaussian Random Process

I have a random process, $$Y_i=Bf_i+W_i,\quad i=0,\ldots,N$$ The RVs $B,W_1,\ldots,W_n$ are i.i.d., with $B\thicksim\mathcal{N}(0,\sigma_b^2)$ and $W_i\thicksim\mathcal{N}(0,\sigma_w^2)$. The ...
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Derivative of a Bivariate normal CDF with respect to its variables

Following up on the question (and answers) here, I'm trying to derive $\frac{\partial \Phi(x_1, x_2|\mathbf{\underline{\theta}})}{\partial x_1}$ and $\frac{\partial \Phi(x_1, x_2|\mathbf{\underline{\...
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Finding conditional expectation $E(Y|X>x)$ of Bivariate Normal distribution

$(X,Y)$ follows bivariate normal distribution with parameters $(\mu_1, \mu_2, \sigma_1^2, \sigma_2^2, \rho)$. Find the conditional expectation $E(Y|X>x)$. I know that $Y|x$ follows Normal ...
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Find $\rho \left( {XY,Y} \right)$ given that $(X,Y)$ follows a bivariate normal distribution

Suppose $X$ and $Y$ have a bivariate normal distribution with parameters ${\mu _X} = {\mu _Y} = 0$, ${\sigma _X}^2 = {\sigma _Y}^2 = 1$, and $\rho = {\rho _{X,Y}} \ne 0$. I'm asked to find the ...
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The $F(b,d)-F(b,c)-F(a,d)+F(a,c)\geqslant 0$ condition of joint CDFs

I'm trying to verify that a certain function of two variables $F(x,y)$ satisfies the conditions of a joint CDF. Showing that each condition holds has been fairly straightforward except, that is, for ...
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Marginal pdf with nonnegative random variables

Suppose that $X_1$ and $X_2$ are two nonnegative random variables that satisfy $$P(X_1 > x_1, X_2 > x_2) = \exp \left[−\mu x_1 − \nu x_2 − \lambda \max(x_1, x_2)\right] ,\text{ for }x_1 \ge 0 \...
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2 votes
1 answer
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Conditional expectation of $X$ given $X+Y=5$ of a bivariate normal distribution

Random variables X and Y have a bivariate normal distribution. If the parameters are $\sigma_x,\sigma_y,\mu_x, \mu_y, \rho$, how do we express $E(X|X+Y=5)$ using those parameters? The conditional ...
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Having trouble grasping bivariate probability distributions

So I'm having trouble grasping bivariate distribution functions. I seem to be struggling with how to determine the upper and lower bounds of my integrals. So for example, this is a question from my ...
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3 votes
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190 views

Joint distribution of two brownian bridges

Consider $X_t=B_t-tB_1$ for $0\leq t\leq 1$, where $B_t$ represents the standard Brownian motion. I derived that $E(X_t)=0$ and $\operatorname{Cov}(X_t,X_s)=\min(t,s)-st$. Now, I am asked the joint ...
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